How to cut a mesh by specifying the angle
It isn't currently supported yet, but still possible. You can say that the angle constrain is based on View coordiante, so let's try a tricky way:


Install Precise Align addon (not a must-have, but very effective here);
Select three vertices, including the specified vertex, click Create Empty button in Precise Align panel in the Tool Shelf (T). Click on Swap Empty Origin button to set it at the specified vertex if it hasn't been there. Also make sure Parent to Object is toggled;
Select the empty, ShiftNumpad 7 to Align its local Top view (Local Z axis towards viewer), thus you can make one adjacent edge visually horizontal (or vertical);
Keep the current point of view, R to rotate the empty along its own pivot point by a specified angle (or the complement/supplementary angle, depending on how you rotate it);
Select the mesh object, enter Edit mode, enable Knife tool, press C to constrain horizontally or vertically based on the view (screen) coordinate. Repeat it if needed.
Delete the empty. The mesh rotation will be regressed automatically.


A demo video for better understanding. It's not perfect, but works.